Supplemental materials for: Rolek, B.W., McClure, CJW, Dunn, L., Curti, M., … Ridgway’s Hawk IPM and PVA

Contact information:

Metadata, data, and scripts used in analyses can be found at https://github.com/The-Peregrine-Fund/XXXXX.

The full workflow below is visible as a html website at: https://the-peregrine-fund.github.io/XXXXX/.

A permanent archive and DOI is available at: https://zenodo.org/doi/XXXXX


Plot model estimates

#load("C:\\Users\\rolek.brian\\OneDrive - The Peregrine Fund\\Documents\\Projects\\Ridgways IPM\\outputs\\ipm_sites.rdata")
load("C:\\Users\\rolek.brian\\OneDrive - The Peregrine Fund\\Documents\\Projects\\Ridgways IPM\\outputs\\ipm1.rdata")
load("data/data.rdata")
library ('MCMCvis')
library ('coda')
library ('ggplot2')
library('reshape2')
out <- list(as.mcmc(post[[1]]), 
             as.mcmc(post[[2]]), 
             as.mcmc(post[[3]]),
             as.mcmc(post[[4]]),
             as.mcmc(post[[5]]) ) #,
             # as.mcmc(post[[6]]),
             # as.mcmc(post[[7]]),
             # as.mcmc(post[[8]]),
             # as.mcmc(post[[9]]),
             # as.mcmc(post[[10]]))

# Identify chains with NAs that 
# failed to initialize
NAlist <- c()
for (i in 1:length(out)){
  NAlist[i] <- any (is.na(out[[i]][,1:286]) | out[[i]][,1:286]<0)
}
# Subset chains to those with good initial values
out <- out[!NAlist]
post2 <- post[!NAlist]
outp <- MCMCpstr(out, type="chains")

!NAlist
## [1] TRUE TRUE TRUE TRUE TRUE
# default settings for plots 
plt  <- function(object, params,...) {
  MCMCplot(object=out, 
           params=params, 
           guide_axis=TRUE, 
           HPD=TRUE, ci=c(80, 95), horiz=FALSE, 
           #ylim=c(-10,10),
           ...)
  }

Plot model estimates of demographic rates. Life Stages are abbreviated as B = breeder, NB = nonbreeder, FY = first year. First-year abundance accounts for translocated birds.

# Abundance of females at Los Haitises
par(mfrow=c(5,2))
plt(object=out, 
    exact=TRUE, ISB=FALSE, 
    params=paste0("NFY[",1:13, ", 1]"), 
    main="First-year (FY) minus hacked\n Los Haitises", 
    labels = 2011:2023,
    xlab = "Year", ylab= "Abundance")
plot(2011:2023, datl$countsFY[,1], 
     ylab="Counts", xlab="Year", type="b")

plt(object=out, 
    exact=TRUE, ISB=FALSE, 
    params=paste0("NF[",1:13, ", 1]"), 
    main="Adult nonbreeder (NB)\n Los Haitises", 
    labels = 2011:2023,
    xlab = "Year", ylab= "Abundance")
plot.new()

plt(object=out, 
    exact=TRUE, ISB=FALSE, 
    params=paste0("NB[",1:13, ", 1]"),
    main="Adult breeder (B)\n Los Haitises", 
    labels = 2011:2023,
    xlab = "Year", ylab= "Abundance")
plot.new()

plt(object=out, 
    exact=TRUE, ISB=FALSE, 
    params=paste0("NAD[",1:13, ", 1]"), 
    main="Adult Breeders and Nonbreeders\n Los Haitises", 
    labels = 2011:2023,
    xlab = "Year", ylab= "Abundance")
plot(2011:2023, datl$countsAdults[,1], 
     ylab="Counts", xlab="Year",  type="b")

plt(object=out, 
    exact=TRUE, ISB=FALSE, 
    params=paste0("Ntot[",1:13, ", 1]"), 
    main="All stages\n Los Haitises", 
    labels = 2011:2023,
    xlab = "Year", ylab= "Abundance")
plot(2011:2023, datl$countsFY[,1]+datl$countsAdults[,1], 
     ylab="Counts", xlab="Year",  type="b")

# Abundance of females at Punta Cana
par(mfrow=c(5,2))
plt(object=out, 
    exact=TRUE, ISB=FALSE, 
    params=paste0("NFY[",1:13, ", 2]"), 
    main="First-year (FY) plus hacked\n Punta Cana", 
    labels = 2011:2023,
    xlab = "Year", ylab= "Abundance")
plot(2011:2023, datl$countsFY[,2], 
     ylab="Counts", xlab="Year",  type="b")

plt(object=out, 
    exact=TRUE, ISB=FALSE, 
    params=paste0("NF[",1:13, ", 2]"), 
    main="Adult nonbreeder (NB)\n Punta Cana", 
    labels = 2011:2023,
    xlab = "Year", ylab= "Abundance")
plot.new()

plt(object=out, 
    exact=TRUE, ISB=FALSE, 
    params=paste0("NB[",1:13, ", 2]"),
    main="Adult breeder (B)\n Punta Cana", 
    labels = 2011:2023,
    xlab = "Year", ylab= "Abundance")
plot.new()

plt(object=out, 
    exact=TRUE, ISB=FALSE, 
    params=paste0("NAD[",1:13, ", 2]"), 
    main="Adult Breeders and Nonbreeders\n Punta Cana", 
    labels = 2011:2023,
    xlab = "Year", ylab= "Abundance")
plot(2011:2023, datl$countsAdults[,2], 
     ylab="Counts", xlab="Year",  type="b")

plt(object=out, 
    exact=TRUE, ISB=FALSE, 
    params=paste0("Ntot[",1:13, ", 2]"), 
    main="All stages\n Punta Cana", 
    labels = 2011:2023,
    xlab = "Year", ylab= "Abundance")
plot(2011:2023, datl$countsFY[,2]+datl$countsAdults[,2], 
     ylab="Counts", xlab="Year",  type="b")

Population dynamics are determined by transitions, These transitions between stages are abbreviated as the starting life stage to the final life stage. For example a first-year recruiting to a breeder would be abbreviated as “FY to B”. I’ll list them here for convenience:

“FY to NB” is first-year to nonbreeder.

“NB to NB” is nonbreeder adult to nonbreeder adult.

“B to NB” is a breeding adult to a nonbreeder adult.

“FY to B” is first-year to breeder.

“NB to B” is nonbreeder adult to breeder adult.

“B to B” is breeder adult to breeder adult.

# Finer population segments
par(mfrow=c(4,2))
plt(object=out, 
    exact=TRUE, ISB=FALSE, 
    params=paste0("N[", 1, ", ", 1:13, ", 1]"), 
    main="Los Haitises\nFirst-years born", 
    labels = 2011:2023, 
    xlab = "Year", ylab= "Abundance")
plt(object=out, 
    exact=TRUE, ISB=FALSE, 
    params=paste0("N[", 2, ", ", 1:13, ", 1]"), 
    main="Los Haitises\nFY to NB", 
    labels = 2011:2023, 
    xlab = "Year", ylab= "Abundance")
plt(object=out, 
    exact=TRUE, ISB=FALSE, 
    params=paste0("N[", 3, ", ", 1:13, ", 1]"), 
    main="Los Haitises\nNB to NB", 
    labels = 2011:2023, 
    xlab = "Year", ylab= "Abundance")
plt(object=out, 
    exact=TRUE, ISB=FALSE, 
    params=paste0("N[", 4, ", ", 1:13, ", 1]"), 
    main="Los Haitises\nB to NB", 
    labels = 2011:2023, 
    xlab = "Year", ylab= "Abundance")
plt(object=out, 
    exact=TRUE, ISB=FALSE, 
    params=paste0("N[", 5, ", ", 1:13, ", 1]"), 
    main="Los Haitises\nFY to B", 
    labels = 2011:2023, 
    xlab = "Year", ylab= "Abundance")
plt(object=out, 
    exact=TRUE, ISB=FALSE, 
    params=paste0("N[", 6, ", ", 1:13, ", 1]"), 
    main="Los Haitises\nNB to B",
    labels = 2011:2023,
    xlab = "Year", ylab= "Abundance")
plt(object=out, 
    exact=TRUE, ISB=FALSE, 
    params=paste0("N[", 7, ", ", 1:13, ", 1]"), 
    main="Los Haitises\nB to B", 
    labels = 2011:2023, 
    xlab = "Year", ylab= "Abundance")


par(mfrow=c(4,2))

plt(object=out, 
    exact=TRUE, ISB=FALSE, 
    params=paste0("N[", 1, ", ", 1:13, ", 2]"), 
    main="Punta Cana\nFY born",
    labels = 2011:2023, 
    xlab = "Year", ylab= "Abundance")
plt(object=out, 
    exact=TRUE, ISB=FALSE, 
    params=paste0("N[", 2, ", ", 1:13, ", 2]"), 
    main="Punta Cana\nFY to NB", 
    labels = 2011:2023, 
    xlab = "Year", ylab= "Abundance")
plt(object=out, 
    exact=TRUE, ISB=FALSE, 
    params=paste0("N[", 3, ", ", 1:13, ", 2]"), 
    main="Punta Cana\nNB to NB", 
    labels = 2011:2023, 
    xlab = "Year", ylab= "Abundance")
plt(object=out, 
    exact=TRUE, ISB=FALSE, 
    params=paste0("N[", 4, ", ", 1:13, ", 2]"), 
    main="Punta Cana\nB to NB", 
    labels = 2011:2023, 
    xlab = "Year", ylab= "Abundance")
plt(object=out, 
    exact=TRUE, ISB=FALSE, 
    params=paste0("N[", 5, ", ", 1:13, ", 2]"), 
    main="Punta Cana\nFY to B", 
    labels = 2011:2023, 
    xlab = "Year", ylab= "Abundance")
plt(object=out, 
    exact=TRUE, ISB=FALSE, 
    params=paste0("N[", 6, ", ", 1:13, ", 2]"), 
    main="Punta Cana\nNB to B",
    labels = 2011:2023, 
    xlab = "Year", ylab= "Abundance")
plt(object=out, 
    exact=TRUE, ISB=FALSE, 
    params=paste0("N[", 7, ", ", 1:13, ", 2]"), 
    main="Punta Cana\nB to B", 
    labels = 2011:2023, 
    xlab = "Year", ylab= "Abundance")

Other parameter estimates.

# I needed to abbreviate to save plot space
# FY=first-year, NB=Nonbreeder, B=Breeder
par(mfrow=c(1,2))
plt(object=out, 
    params=paste0("mus[",1:8, ", 1]"), 
    exact=TRUE, ISB=FALSE, 
    ylim=c(0,1),
    main="Overall means\n Los Haitises", 
    labels=c("FY survival", "NB survival", "B survival",
             "FY to B", "NB to B", "B to NB",
             "NB detection", "B detection")
    )

plt(object=out, 
    params=paste0("mus[",1:8, ", 2]"), 
    exact=TRUE, ISB=FALSE, 
    ylim=c(0,1),
    main="Overall means\n Punta Cana", 
    labels=c("FY survival", "NB survival", "B survival",
             "FY to B", "NB to B", "B to NB",
             "NB detection", "B detection"))

par(mfrow=c(1,1))
plt(object=out, 
    params="betas", 
    main= "Translocation effects",
    labels=c("FY survival", "NB survival", "B survival",
             "FY to B", "NB to B", "B to NB",
             "NB detection", "B detection"))

# Get predicted survival, recruitment, and detection
# with effects from tranlsocation and hacking
# Birds were only hacked from LHNP to PC here
# so we only predict values for PC
pred.mus <- array(NA, dim=c(dim(outp$mus)))
for (m in c(1,2,3,5,7)){
    for (tr in 1:2){
    pred.mus[m,tr,] <- plogis( outp$lmus[m,2,] + outp$betas[m,]*c(0,1)[tr] ) 
    }}
# treatment, mu, site, iter
mus.md <- apply(pred.mus, c(1,2), median)
mus.HDI80 <- apply(pred.mus, c(1,2), HDInterval::hdi, credMass=0.8)
mus.HDI95 <- apply(pred.mus, c(1,2), HDInterval::hdi)
par(mar=c(8,5,1,1), mfrow=c(1,1))
for (tr in 1:2){
  if (tr==1){
plot(1:5+c(-0.1,0.1)[tr], mus.md[c(1,2,3,5,7),tr], 
     pch=c(16, 17)[tr], cex=2,
     ylim=c(0,1),
     xlim=c(0.5, 5.5), 
     cex.axis=2, cex.lab=2,
     ylab="Probability", xlab="",
     main="",
     xaxt="n", yaxt="n")}
  else{
    points(1:5+c(-0.1,0.1)[tr], mus.md[c(1,2,3,5,7),tr], 
         pch=c(16, 17)[tr], cex=2)
  }}

axis(1, at=1:5, cex.axis=1.5,
     labels=c("Survival\nFY", "Survival\nNB", "Survival\nB", 
               "Recruitment\nNB to B", 
              "Detection\nNB"), las=2)
axis(2, at=c(0, 0.25, 0.5, 0.75, 1), cex.axis=2,
     labels=c(0, NA, 0.5, NA, 1))
for (m in 1:5){
for (tr in 1:2){
lines(rep(c(1:5)[m] + c(-0.1,0.1)[tr],2), mus.HDI95[1:2,c(1,2,3,5,7)[m],tr], lwd=3)
lines(rep(c(1:5)[m] + c(-0.1,0.1)[tr],2), mus.HDI80[1:2,c(1,2,3,5,7)[m],tr], lwd=6)
}}
legend(x=1.15,y=0.4, pch=c(16,17), pt.cex=2, cex=1.5, xpd=NA, horiz=T, 
       legend=c("Unmanaged", "Translocated" ))

# Fecundity
par(mfrow=c(1,1))
plt(object=out, 
    params=c("lmu.f"), 
    labels= c("Fecundity\n(log scale)\nLos Haitises",
              "Fecundity\n(log scale)\nPunta Cana"))

f <- exp(outp$lmu.f)
f.md <- apply(f, 1, median)
f.HDI80 <- apply(f, 1, HDInterval::hdi, credMass=0.8)
f.HDI95 <- apply(f, 1, HDInterval::hdi)

# Calculate treatment effects on fecundity
f.pred <- array(NA, dim=dim(outp$lmu.f))
for (s in 1:2){
  f.pred[s,] <- exp(outp$lmu.f[s,] + outp$gamma[,1])
} # s
f2.md <- apply(f.pred, 1, median)
f2.HDI80 <- apply(f.pred, 1, HDInterval::hdi, credMass=0.8)
f2.HDI95 <- apply(f.pred, 1, HDInterval::hdi)

par(mar=c(6,5,1,1))
plot(c(1, 2)-0.1, f.md, 
     pch=16, cex=3,
     ylim=c(min(f.HDI95), max(f2.HDI95)),
     xlim=c(0.5, 2.5), 
     ylab="Fecundity", xlab="",
     main="",
     cex.axis=2, cex.lab=2,
     xaxt="n", yaxt="n")
axis(1, at=c(1,2), cex.axis=2,
     labels=c("Los Haitises\nNational Park","Punta Cana"),
     padj=1)
axis(2, at=c(0, 0.2, 0.4, 0.6, 0.8), cex.axis=2,
     labels=c(0, NA, 0.4, NA, 0.8))
lines(c(1,1)-0.1, f.HDI95[,1], lwd=3)
lines(c(2,2)-0.1, f.HDI95[,2], lwd=3)
lines(c(1,1)-0.1, f.HDI80[,1], lwd=6)
lines(c(2,2)-0.1, f.HDI80[,2], lwd=6)

points(c(1.1, 2.1), f2.md, 
     pch=17, cex=3)

lines(c(1,1) +0.1, f2.HDI95[,1], lwd=3)
lines(c(2,2) +0.1, f2.HDI95[,2], lwd=3)
lines(c(1,1) +0.1, f2.HDI80[,1], lwd=6)
lines(c(2,2) +0.1, f2.HDI80[,2], lwd=6)
legend(x=1.9,y=0.8, pch=c(16,17), pt.cex=3, cex=1.5,
       legend=c("Untreated", "Treated" ) )

# Is fecundity at LHNP greater than PC
par(mfrow=c(1,1))
fdiff <- f[1,]-f[2,]
hist(fdiff, main="Fecundity difference\nbetween untreated sites")

# pd=
mean(fdiff>0)
## [1] 0.9787
# How big are the treatment effects on fecundity at each site
median(f.pred[1,]/f[1,])
## [1] 6.147469
median(f.pred[2,]/f[2,])
## [1] 6.147469
# gamma = nest treatment effect on fecundity
par(mfrow=c(1,1))
plt(object=out, 
    params=c("gamma"), 
    main="Anti-Parasitic Fly\nTreatment Effects", ylim=c(0,3))

par(mfrow=c(1,1))
sds <- paste0("sds[", 1:9, "]")
plt(object=out, params=sds,
    exact=TRUE, ISB=FALSE,
    main="Temporal SDs (synchrony among sites)", 
    labels=c("FY survival", "NB survival", "B survival",
             "FY to B", "NB to B", "B to NB",
             "NB detection", "B detection",
             "Fecundity"))

sds2 <- paste0("sds2[", 1:9, "]")
plt(object=out, params=sds2,
    exact=TRUE, ISB=FALSE,
    main="Site-temporal SDs", 
    labels=c("FY survival", "NB survival", "B survival",
             "FY to B", "NB to B", "B to NB",
             "NB detection", "B detection",
             "Fecundity"))

# Correlations among vital rates
# Plot is messy with only a few strong correlations
ind <- 1
Rs <- R2s <- c()
for (i in 1:(nrow(outp$R)-1)){
  for (j in (i+1):nrow(outp$R)){
  Rs[ind] <- paste0("R[",i,", ", j, "]")
  R2s[ind] <- paste0("R2[",i,", ", j, "]")
  ind <- ind+1
  }}
par(mfrow=c(2,1))
plt(object=out, params=Rs[1:18], exact=TRUE, ISB=FALSE, 
    main="Correlations btw demographic rates\n over time (synchrony)",
    xlab = "Rhos", guide_lines=TRUE)
plt(object=out, params=Rs[19:36], exact=TRUE, ISB=FALSE, 
    main="Correlations btw demographic rates\n over time (synchrony), continued...",
    xlab = "Rhos", guide_lines=TRUE)

par(mfrow=c(2,1))
plt(object=out, params=R2s[1:18], exact=TRUE, ISB=FALSE, 
    main="Correlations btw demographic rates\n over time and sites",
    xlab = "Rhos", guide_lines=TRUE)
plt(object=out, params=R2s[19:36], exact=TRUE, ISB=FALSE, 
    main="Correlations btw demographic rates\n over time and sites, continued ...",
    xlab = "Rhos", guide_lines=TRUE)

# Annual averages for integration into the population model
par(mfrow=c(1,1))
labs <- c(paste0("LH ",2011:2023), paste0("PC ",2011:2023))
plt(object=out, params="mn.phiFY", ylim=c(0,1),
    main="First-year survival", labels = labs,
    xlab = "Year", ylab= "Survival")
abline(v=13.5, lwd=2)

plt(object=out, params="mn.phiA", ylim=c(0,1),
    main="Adult nonbreeder", labels = labs,
    xlab = "Year", ylab= "Survival")
abline(v=13.5, lwd=2)

plt(object=out, params="mn.phiB", ylim=c(0,1),
    main="Breeder", labels = labs,
    xlab = "Year", ylab= "Survival")
abline(v=13.5, lwd=2)

plt(object=out, params="mn.psiFYB", ylim=c(0,1),
    main="First-year to breeder", labels = labs,
    xlab = "Year", ylab= "Recruitment")
abline(v=13.5, lwd=2)

plt(object=out, params="mn.psiAB", ylim=c(0,1),
    main="Adult nonbreeder to breeder", labels = labs,
    xlab = "Year", ylab= "Recruitment")
abline(v=13.5, lwd=2)

plt(object=out, params="mn.psiBA", ylim=c(0,1),
    main="Adult breeder to nonbreeder", labels = labs,
    xlab = "Year", ylab= "Recruitment")
abline(v=13.5, lwd=2)

plt(object=out, params="mn.pA", ylim=c(0,1),
    main="Nonbreeder", labels = labs,
    xlab = "Year", ylab= "Detection")
abline(v=13.5, lwd=2)

plt(object=out, params="mn.pB", ylim=c(0,1),
    main="Breeder", labels = labs,
    xlab = "Year", ylab= "Detection")
abline(v=13.5, lwd=2)

plt(object=out, params="mn.f",
    main="", labels=labs,
    xlab = "Year", ylab= "Fecundity")
abline(v=13.5, lwd=2)

Stage Structure of Each Population

mdFY <-  apply(outp$NFY, c(1,2), median) 
mdB <-  apply(outp$NB, c(1,2), median) 
mdF <-  apply(outp$NF, c(1,2), median) 
lFY <- melt(mdFY)
lB <- melt(mdB)
lF <- melt(mdF)
lFY$Stage <- "First-year"
lB$Stage <- "Breeder"
lF$Stage <- "Nonbreeder"
ldat <- rbind(lFY, lB, lF)
colnames(ldat)[1:3] <- c("Year", "Sitenum", "Number") 
ldat$Site <- ifelse(ldat$Sitenum==1, "Los Haitises", "Punta Cana")

# Use median number of females in each stage
# to plot an approximate population structure
ggplot(ldat, aes(fill=Stage, y=as.numeric(Number), x=Year)) + 
  geom_bar(position="fill", stat="identity") +
  ylab("Proportion of population") + 
  facet_wrap("Site")

ggplot(ldat, aes(fill=Stage, y=as.numeric(Number), x=Year)) + 
  geom_bar(position="stack", stat="identity") +
  ylab("Numer of females") + 
  facet_wrap("Site", scales = "free_y")

Model diagnostics

Check Goodness-of-fit

Goodness-of-fit tests provide evidence that statistical distributions adequately describe the data. Here we test fit for brood size and counts. A Bayesian p-value nearest to 0.5 suggests good fitting statistical distributions, while values near 1 or 0 suggest poor fit.

# Goodness of fit check
fit.check <- function(out, ratio=FALSE, 
                      name.rep="f.dmape.rep", 
                      name.obs="f.dmape.obs",
                      jit=100,
                      ind=1,
                      lab=""){
  par(mfrow=c(1,1))
  # plot mean absolute percentage error
  samps <- MCMCpstr(out, "all", type="chains")
  rep <- samps[name.rep][[1]][ind,]
  obs <- samps[name.obs][[1]][ind,]
  mx <- max(c(rep, obs))
  mn <- min(c(rep, obs))
  plot(jitter(obs, amount=jit), 
       jitter(rep, amount=jit),
       main=paste0("Mean absolute percentage error\n",lab),
       ylab="Discrepancy replicate values",
       xlab="Discrepancy observed values", 
       xlim=c(mn,mx), ylim=c(mn,mx), 
       pch=16, cex=0.5, col="gray10")
  curve(1*x, from=mn, to=mx, add=T, lty=2, lwd=2, col="blue")
  bp1 <- round(mean(rep > obs),2)
  loc <- ifelse(bp1 < 0.5, "topleft", "bottomright")
  legend(loc, legend=bquote(p[B]==.(bp1)), bty="n", cex=2)
  
  if (ratio==TRUE){
    t.rep <- samps["tvm.rep"][[1]][ind,]
    t.obs <- samps["tvm.obs"][[1]][ind,]
    # plot variance/mean ratio
    hist(t.rep, nclass=50,
         xlab="variance/mean ", main=NA, axes=FALSE)
    abline(v=t.obs, col="red")
    axis(1); axis(2)
  }
  return(list('Bayesian p-value'=bp1))
}

# check goodness-of-fit for brood size
# breeder, ind=1
# fit.check(out, ratio=F,
#           name.rep="dmape.rep", 
#           name.obs="dmape.obs",
#           ind=1,
#           lab="Breeder counts- Poisson", jit=300)
# # nonbreeder, ind=2
# fit.check(out, ratio=F,
#           name.rep="dmape.rep", 
#           name.obs="dmape.obs",
#           ind=2,
#           lab="Nonbreeder counts- Poisson", jit=300)

fit.check(out, ratio=F,
          name.rep="dmape.rep", 
          name.obs="dmape.obs",
          ind=1,
          lab="Adults(Breeder+Nonbreeder)- Poisson", jit=300)

## $`Bayesian p-value`
## [1] 0.04
# first-year, ind=2
# poisson failed fit test bp=0
# Currently running models to try and fix
fit.check(out, ratio=F,
          name.rep="dmape.rep", 
          name.obs="dmape.obs",
          ind=2,
          lab="First-year counts\nNeg binomial-Poisson", jit=300)

## $`Bayesian p-value`
## [1] 0.38
# fecundity
fit.check(out, ratio=F,
          name.rep="f.dmape.rep", 
          name.obs="f.dmape.obs",
          ind=1,
          lab="Fecundity-Neg binomial", jit=300)

## $`Bayesian p-value`
## [1] 0.81

Examine Traceplots

Traceplots provide evidence that models adequately converged.

MCMCtrace(post2, pdf=FALSE, params= "sds")

MCMCtrace(post2, pdf=FALSE, params= "sds2")

MCMCtrace(post2, pdf=FALSE, params= "mus")

MCMCtrace(post2, pdf=FALSE, params= "betas")

MCMCtrace(post2, pdf=FALSE, params= "NF")

MCMCtrace(post2, pdf=FALSE, params= "NFY")

MCMCtrace(post2, pdf=FALSE, params= "NB")

MCMCtrace(post2, pdf=FALSE, params= "R")